{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "colab_type": "text",
        "id": "view-in-github"
      },
      "source": [
        "\u003ca href=\"https://colab.research.google.com/github/google/neural-tangents/blob/main/notebooks/phase_diagram.ipynb\" target=\"_parent\"\u003e\u003cimg src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/\u003e\u003c/a\u003e"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "colab_type": "text",
        "id": "cIkq2O7yVWYJ"
      },
      "source": [
        "Copyright 2020 Google LLC\n",
        "\n",
        "Licensed under the Apache License, Version 2.0 (the \"License\");\n",
        "you may not use this file except in compliance with the License.\n",
        "You may obtain a copy of the License at\n",
        "\n",
        "     https://www.apache.org/licenses/LICENSE-2.0\n",
        "\n",
        "Unless required by applicable law or agreed to in writing, software\n",
        "distributed under the License is distributed on an \"AS IS\" BASIS,\n",
        "WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n",
        "See the License for the specific language governing permissions and\n",
        "limitations under the License."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "colab_type": "text",
        "id": "aWMU1YkS5lPB"
      },
      "source": [
        "### Imports \u0026 Utils"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 0,
      "metadata": {
        "colab": {
          "height": 34
        },
        "colab_type": "code",
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          "elapsed": 380,
          "status": "ok",
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        "id": "-U2XtTJ-kma4",
        "outputId": "258648be-06e5-41d1-d6d2-c9f18ccf7d45"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "/bin/sh: pip: command not found\n"
          ]
        }
      ],
      "source": [
        "!pip install -q --upgrade pip\n",
        "!pip install -q --upgrade jax[cuda11_cudnn82] -f https://storage.googleapis.com/jax-releases/jax_cuda_releases.html\n",
        "!pip install -q git+https://www.github.com/google/neural-tangents"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 0,
      "metadata": {
        "colab": {},
        "colab_type": "code",
        "id": "-1nTu8wtiK5w"
      },
      "outputs": [],
      "source": [
        "import jax.numpy as jnp\n",
        "\n",
        "from jax.example_libraries import optimizers\n",
        "from jax import grad, jit, vmap\n",
        "from jax import lax\n",
        "from jax import config\n",
        "config.update('jax_enable_x64', True)\n",
        "\n",
        "from functools import partial\n",
        "\n",
        "import neural_tangents as nt\n",
        "from neural_tangents import stax\n",
        "\n",
        "_Kernel = nt._src.utils.kernel.Kernel\n",
        "\n",
        "def Kernel(K):\n",
        "  \"\"\"Create an input Kernel object out of an jnp.ndarray.\"\"\"\n",
        "  return _Kernel(cov1=jnp.diag(K), nngp=K, cov2=None, \n",
        "                 ntk=None, is_gaussian=True, is_reversed=False,\n",
        "                 diagonal_batch=True, diagonal_spatial=False,\n",
        "                 shape1=(K.shape[0], 1024), shape2=(K.shape[1], 1024),\n",
        "                 x1_is_x2=True, is_input=True, batch_axis=0, channel_axis=1,\n",
        "                 mask1=None, mask2=None) \n",
        "  \n",
        "def fixed_point(f, initial_value, threshold):\n",
        "  \"\"\"Find fixed-points of a function f:R-\u003eR using Newton's method.\"\"\"\n",
        "  g = lambda x: f(x) - x\n",
        "  dg = grad(g)\n",
        "\n",
        "  def cond_fn(x):\n",
        "    x, last_x = x\n",
        "    return jnp.abs(x - last_x) \u003e threshold\n",
        "\n",
        "  def body_fn(x):\n",
        "    x, _ = x\n",
        "    return x - g(x) / dg(x), x\n",
        "  \n",
        "  return lax.while_loop(cond_fn, body_fn, (initial_value, 0.0))[0]"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 0,
      "metadata": {
        "colab": {},
        "colab_type": "code",
        "id": "xjtNH1OL7vl8"
      },
      "outputs": [],
      "source": [
        "from IPython.display import set_matplotlib_formats\n",
        "set_matplotlib_formats('pdf', 'svg')\n",
        "import matplotlib.pyplot as plt\n",
        "import seaborn as sns\n",
        "sns.set_style(style='white')\n",
        "\n",
        "def format_plot(x='', y='', grid=True):  \n",
        "  ax = plt.gca()\n",
        "    \n",
        "  plt.grid(grid)\n",
        "  plt.xlabel(x, fontsize=20)\n",
        "  plt.ylabel(y, fontsize=20)\n",
        "  \n",
        "def finalize_plot(shape=(1, 1)):\n",
        "  plt.gcf().set_size_inches(\n",
        "    shape[0] * 1.5 * plt.gcf().get_size_inches()[1], \n",
        "    shape[1] * 1.5 * plt.gcf().get_size_inches()[1])\n",
        "  plt.tight_layout()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "colab_type": "text",
        "id": "MF_hLTxuiHbD"
      },
      "source": [
        "# Phase Diagram"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "colab_type": "text",
        "id": "h4T7uqCL7r5e"
      },
      "source": [
        "We will reproduce the phase diagram described in [Poole et al.](https://papers.nips.cc/paper/6322-exponential-expressivity-in-deep-neural-networks-through-transient-chaos) and [Schoenholz et al.](https://arxiv.org/abs/1611.01232) using Neural Tangents. In these and subsequent papers, it was found that deep neural networks can exhibit a phase transition as a function of the variance of their weights ($\\sigma_w^2$) and biases ($\\sigma_b^2$). For networks with $\\tanh$ activation functions, this phase transition is between an \"ordered\" phase and a \"chaotic\" phase. In the ordered phase, pairs of inputs collapse to a single point as they propagate through the network. By contrast, in the chaotic phase, nearby inputs become increasingly dissimilar in later layers of the network. This phase diagram is shown below. A number of properties of neural networks - such as trainability, mode-collapse, and maximum learing rate - have now been related to this phase diagram over many papers (recently e.g. [Yang et al.](https://arxiv.org/abs/1902.08129), [Jacot et al.](https://arxiv.org/abs/1907.05715), [Hayou et al.](https://arxiv.org/abs/1905.13654), and [Xiao et al.](https://arxiv.org/abs/1912.13053)).\n",
        "\n",
        "\\\n",
        "\n",
        "![Phase Diagram](https://raw.githubusercontent.com/google/neural-tangents/main/notebooks/figures/pennington_phase_diagram.svg?sanitize=true)\n",
        "\n",
        "\u003e Phase diagram for $\\tanh$ neural networks (appeared in [Pennington et al.](https://arxiv.org/abs/1802.09979)).\n",
        "\n",
        "\\\n",
        "\n",
        "Consider two inputs to a neural network, $x_1$ and $x_2$, normalized such that $\\|x_1\\| = \\|x_2\\| = q^0$. We can compute the cosine-angle between the inputs, $c^0 = \\cos\\theta_{12} = \\frac{x_1 \\cdot x_2}{q^0}$. Additionally, we can keep track of the norm and cosine angle of the resulting pre-activations ($q^l$ and $c^l$ respectively) as signal passes through layers of the neural network. In the wide-network limit there are deterministic functions, called the $\\mathcal Q$-map and the $\\mathcal{C}$-map, such that $q^{l+1} = \\mathcal Q(q^l)$ and $c^{l+1} = \\mathcal C(q^l, c^l)$. \n",
        "\n",
        "\\\n",
        "\n",
        "In fully-connected networks with $\\tanh$-like activation functions, both the $\\mathcal Q$-map and $\\mathcal C$-map have unique stable-fixed-points, $q^*$ and $c^*$, such that $q^* = \\mathcal Q(q^*)$ and $c^* = \\mathcal C(q^*, c^*)$. To simplify the discussion, we typically choose to normalize our inputs so that $q^0 = q^*$ and we can restrict our study to the $\\mathcal C$-map. The $\\mathcal C$-map always has a fixed point at $c^* = 1$ since two identical inputs will remain identical as they pass through the network. However, this fixed point is not always stable and two points that start out very close together will often separate. Indeed, the ordered and chaotic phases are characterized by the stability of the $c^* = 1$ fixed point. In the ordered phase $c^* = 1$ is stable and pairs of inputs converge to one another as they pass through the network. In the chaotic phase the $c^* = 1$ point is unstable and a new, stable, fixed point with $c^* \u003c 1$ emerges. The phase boundary is defined as the point where $c^* = 1$ is marginally stable.\n",
        "\n",
        "\\\n",
        "\n",
        "To understand the stability of a fixed point, $c^*$, we will use the standard technique in Dynamical Systems theory and expand the $\\mathcal C$-map in $\\epsilon^l = c^l - c^*$ which implies that $\\epsilon^{l+1} = \\chi(c^*)\\epsilon^l$ where $\\chi = \\frac{\\partial\\mathcal C}{\\partial C}$. This implies that sufficiently close to a fixed point of the dynamics, $\\epsilon^l = \\chi(c^*)^l$. If $\\chi(c^*) \u003c 1$ then the fixed point is stable and points move towards the fixed point exponentially quickly. If $\\chi(c^*) \u003e 1$ then points move away from the fixed point exponentially quickly. This implies that the phase boundary, being defined by the marginal stability of $c^* = 1$, will be where $\\chi_1 = \\chi(1) = 1$.  \n",
        "\n",
        "\\\n",
        "\n",
        "To reproduce these results in Neural Tangents, we notice first that the $\\mathcal{C}$-map described above is intimately related to the NNGP kernel, $K^l$, of [Lee et al.](https://arxiv.org/abs/1711.00165), [Matthews et al.](https://arxiv.org/abs/1804.11271), and [Novak et al.](https://arxiv.org/abs/1810.05148). The core of Neural Tangents is a map $\\mathcal T$ for a wide range of architectures such that $K^{l + 1} = \\mathcal T(K^l)$. Since $C^l$ can be written in terms of the NNGP kernel as $C^l = K^l_{12} / q^*$ this implies that Neural Tangents provides a way of computing the $\\mathcal{C}$-map for a wide range of network architectures.\n",
        "\n",
        "\\\n",
        "\n",
        "To produce the phase diagam above, we must compute $q^*$ and $c^*$ as well as $\\chi_1$. We will use a fully-connected network with $\\text{Erf}$ activation functions since they admit an analytic kernel function and are very similar to $\\tanh$ networks. We will first define the $\\mathcal Q$-map by noting that the $\\mathcal Q$-map will be identical to $\\mathcal T$ if the covariance matrix has only a single entry. We will use Newton's method to find $q^*$ given the $\\mathcal Q$-map. Next we will use the relationship above to define the $\\mathcal C$-map in terms of $\\mathcal T$. We will again use Newton's method to find the stable $c^*$ fixed point. We can define $\\chi$ by using JAX's automatic differentiation to compute the derivative of the $\\mathcal C$-map. This can be written relatively concisely below.\n",
        "\n",
        "\\\n",
        "\n",
        "Note: this particular phase diagram holds for a wide range of neural networks but, emphatically, not for ReLUs. The ReLU phase diagram is somewhat different and could be investigated using Neural Tangents. However, we will save it for a followup notebook."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 0,
      "metadata": {
        "colab": {},
        "colab_type": "code",
        "id": "iNS7Qrb7LHY2"
      },
      "outputs": [],
      "source": [
        "def c_map(W_var, b_var):\n",
        "  W_std = jnp.sqrt(W_var)\n",
        "  b_std = jnp.sqrt(b_var)\n",
        "\n",
        "  # Create a single layer of a network as an affine transformation composed\n",
        "  # with an Erf nonlinearity.\n",
        "  kernel_fn = stax.serial(stax.Dense(1024, W_std, b_std), stax.Erf())[2]\n",
        "\n",
        "  def q_map_fn(q):\n",
        "    return kernel_fn(Kernel(jnp.array([[q]]))).nngp[0, 0]\n",
        "  \n",
        "  qstar = fixed_point(q_map_fn, 1.0, 1e-7)\n",
        "\n",
        "  def c_map_fn(c):\n",
        "    K = jnp.array([[qstar, qstar * c], [qstar * c, qstar]])\n",
        "    K_out = kernel_fn(Kernel(K)).nngp\n",
        "    return K_out[1, 0] / qstar\n",
        "\n",
        "  return c_map_fn\n",
        "\n",
        "c_star = lambda W_var, b_var: fixed_point(c_map(W_var, b_var), 0.1, 1e-7)\n",
        "chi = lambda c, W_var, b_var: grad(c_map(W_var, b_var))(c)\n",
        "chi_1 = partial(chi, 1.)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "colab_type": "text",
        "id": "wlZlJGUQEg00"
      },
      "source": [
        "To generate the phase diagram above, we would like to compute the fixed-point correlation not only at a single value of $(\\sigma_w^2,\\sigma_b^2)$ but on a whole mesh. We can use JAX's `vmap` functionality to do this. Here we define vectorized versions of the above functions."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 0,
      "metadata": {
        "colab": {},
        "colab_type": "code",
        "id": "zyZhNMBnUJbC"
      },
      "outputs": [],
      "source": [
        "def vectorize_over_sw_sb(fn):\n",
        "  # Vectorize over the weight variance.\n",
        "  fn = vmap(fn, (0, None))\n",
        "  # Vectorize over the bias variance.\n",
        "  fn = vmap(fn, (None, 0))\n",
        "\n",
        "  return fn\n",
        "\n",
        "c_star = jit(vectorize_over_sw_sb(c_star))\n",
        "chi_1 = jit(vectorize_over_sw_sb(chi_1))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "colab_type": "text",
        "id": "WSFKfaItXPkz"
      },
      "source": [
        "We can use these functions to plot $c^*$ as a function of the weight and bias variance. As expected, we see a region where $c^* = 1$ and a region where $c^* \u003c 1$."
      ]
    },
    {
      "cell_type": "code",
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        "colab_type": "code",
        "executionInfo": {
          "elapsed": 1139,
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          "timestamp": 1586275465411,
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        "id": "V9x74oakFjSk",
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      },
      "outputs": [
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367.788 \nL 216.436472 367.788 \nL 217.545748 367.788 \nL 218.655024 367.788 \nL 219.764301 367.788 \nL 220.239705 367.788 \nL 220.873577 367.788 \nL 221.556208 367.788 \nL 221.982853 367.788 \nL 222.111839 367.788 \nL 223.092129 367.788 \nL 223.677581 367.788 \nL 224.201406 367.788 \nL 225.310682 367.788 \nL 226.419958 367.788 \nL 227.113256 367.788 \nL 227.529234 367.788 \nL 227.945213 367.788 \nL 228.638511 367.788 \nL 229.747787 367.788 \nL 230.515747 367.788 \nL 230.857063 367.788 \nL 231.966339 367.788 \nL 233.075616 367.788 \nL 234.184892 367.788 \nL 235.294168 367.788 \nL 235.965572 367.788 \nL 236.403444 367.788 \nL 237.235402 367.788 \nL 237.512721 367.788 \nL 237.820853 367.788 \nL 238.621997 367.788 \nL 239.652039 367.788 \nL 239.731273 367.788 \nL 240.840549 367.788 \nL 241.042236 367.788 \nL 241.949826 367.788 \nL 243.059102 367.788 \nL 244.168378 367.788 \nL 244.80225 367.788 \nL 245.277654 367.788 \nL 245.567031 367.788 \nL 246.386931 367.788 \nL 247.098542 367.788 \nL 247.496207 367.788 \nL 248.454218 367.788 \nL 248.605483 367.788 \nL 249.714759 367.788 \nL 250.094249 367.788 \nL 250.824036 367.788 \nL 251.573547 367.788 \nL 251.933312 367.788 \nL 252.633907 367.788 \nL 253.042588 367.788 \nL 254.151864 367.788 \nL 254.447671 367.788 \nL 255.261141 367.788 \nL 256.370417 367.788 \nL 257.479693 367.788 \nL 258.588969 367.788 \nL 259.698246 367.788 \nL 260.501515 367.788 \nL 260.807522 367.788 \nL 261.547039 367.788 \nL 261.916798 367.788 \nL 263.026074 367.788 \nL 264.135351 367.788 \nL 265.244627 367.788 \nL 266.353903 367.788 \nL 267.46318 367.788 \nL 268.449203 367.788 \nL 268.572456 367.788 \nL 269.681732 367.788 \nL 270.791008 367.788 \nL 271.160767 367.788 \nL 271.900285 367.788 \nL 273.009561 367.788 \nL 274.118837 367.788 \nL 275.228113 367.788 \nL 276.33739 367.788 \nL 277.446666 367.788 \nL 278.555942 367.788 \nL 279.665218 367.788 \nL 280.169435 367.788 \nL 280.774495 367.788 \nL 281.286468 367.788 \nL 281.883771 367.788 \nL 282.993047 367.788 \nL 284.102323 367.788 \nL 285.2116 367.788 \nL 286.064889 367.788 \nL 286.320876 367.788 \nL 287.430152 367.788 \nL 288.247514 367.788 \nL 288.539428 367.788 \nL 289.648705 367.788 \nL 290.250883 367.788 \nL 290.757981 367.788 \nL 291.867257 367.788 \nL 292.255504 367.788 \nL 292.976533 367.788 \nL 293.971057 367.788 \nL 294.08581 367.788 \nL 295.195086 367.788 \nL 295.934603 367.788 \nL 296.304362 367.788 \nL 296.859 367.788 \nL 297.413638 367.788 \nL 297.690957 367.788 \nL 298.522915 367.788 \nL 299.150807 367.788 \nL 299.632191 367.788 \nL 300.741467 367.788 \nL 301.850743 367.788 \nL 302.294454 367.788 \nL 302.96002 367.788 \nL 303.310317 367.788 \nL 304.069296 367.788 \nL 304.724777 367.788 \nL 305.178572 367.788 \nL 306.176921 367.788 \nL 306.287848 367.788 \nL 307.397125 367.788 \nL 308.506401 367.788 \nL 309.477018 367.788 \nL 309.615677 367.788 \nL 310.724953 367.788 \nL 311.83423 367.788 \nL 312.943506 367.788 \nL 314.052782 367.788 \nL 315.162058 367.788 \nL 316.271335 367.788 \nL 317.380611 367.788 \nL 317.79659 367.788 \nL 318.489887 367.788 \nL 319.183185 367.788 \nL 319.599163 367.788 \nL 320.70844 367.788 \nL 321.817716 367.788 \nL 322.926992 367.788 \nL 323.229522 367.788 \nL 324.036268 367.788 \nL 325.145545 367.788 \nL 326.254821 367.788 \nL 327.364097 367.788 \nL 327.692772 367.788 \nL 328.473373 367.788 \nL 329.351551 367.788 \nL 329.58265 367.788 \nL 330.37499 367.788 \nL 330.691926 367.788 \nL 330.815179 367.788 \nL 331.801202 367.788 \nL 332.910479 367.788 \nL 334.019755 367.788 \nL 334.280761 367.788 \nL 335.129031 367.788 \nL 336.238307 367.788 \nL 337.347584 367.788 \nL 337.606415 367.788 \nL 338.45686 367.788 \nL 339.254152 367.788 \nL 339.566136 367.788 \nL 340.675412 367.788 \nL 341.23005 367.788 \nL 341.784689 367.788 \nL 342.67211 367.788 \nL 342.893965 367.788 \nL 344.003241 367.788 \nL 345.112517 367.788 \nL 346.063326 367.788 \nL 346.221794 367.788 \nL 346.380262 367.788 \nL 347.33107 367.788 \nL 348.440346 367.788 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367.788 \nL 384.454848 367.788 \nL 385.046462 367.788 \nL 386.155739 367.788 \nL 387.265015 367.788 \nL 388.004532 367.788 \nL 388.374291 367.788 \nL 389.483567 367.788 \nL 390.592844 367.788 \nL 391.70212 367.788 \nL 392.224132 367.788 \nL 392.811396 367.788 \nL 393.920672 367.788 \nL 395.029949 367.788 \nL 395.584587 367.788 \nL 396.139225 367.788 \nL 396.139225 366.415116 \nL 396.139225 365.042233 \nL 396.139225 363.669349 \nL 396.139225 362.296466 \nL 396.139225 360.923582 \nL 396.139225 359.550699 \nL 396.139225 358.177815 \nL 396.139225 356.804932 \nL 396.139225 355.432048 \nL 396.139225 354.059165 \nL 396.139225 352.686281 \nL 396.139225 351.313398 \nL 396.139225 349.940514 \nL 396.139225 348.567631 \nL 396.139225 347.194747 \nL 396.139225 345.821863 \nL 396.139225 344.44898 \nL 396.139225 343.076096 \nL 396.139225 341.703213 \nL 396.139225 340.330329 \nL 396.139225 338.957446 \nL 396.139225 337.584562 \nL 396.139225 336.211679 \nL 396.139225 334.838795 \nL 396.139225 333.465912 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315.600238 \nL 379.458004 315.618426 \nL 378.390805 316.073081 \nL 377.281529 316.545547 \nL 376.234839 316.991309 \nL 376.172252 317.017573 \nL 375.062976 317.485697 \nL 373.9537 317.953578 \nL 372.980496 318.364193 \nL 372.844424 318.420745 \nL 371.735147 318.884196 \nL 370.625871 319.347393 \nL 369.693154 319.737076 \nL 369.516595 319.809711 \nL 368.407319 320.268387 \nL 367.298042 320.726797 \nL 366.371379 321.10996 \nL 366.188766 321.184283 \nL 365.07949 321.638078 \nL 363.970214 322.091594 \nL 363.01359 322.482843 \nL 362.860937 322.544273 \nL 361.751661 322.993077 \nL 360.642385 323.441588 \nL 359.618031 323.855727 \nL 359.533109 323.889495 \nL 358.423832 324.333193 \nL 357.314556 324.776583 \nL 356.20528 325.219676 \nL 356.183057 325.22861 \nL 355.096004 325.658236 \nL 353.986727 326.096384 \nL 352.877451 326.53422 \nL 352.707966 326.601494 \nL 351.768175 326.968019 \nL 350.658899 327.400802 \nL 349.549622 327.833254 \nL 349.189218 327.974378 \nL 348.440346 328.262361 \nL 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340.330329 \nL 314.052782 340.671303 \nL 312.943506 341.040031 \nL 311.83423 341.408225 \nL 310.946001 341.703213 \nL 310.724953 341.774777 \nL 309.615677 342.13622 \nL 308.506401 342.497089 \nL 307.397125 342.857401 \nL 306.725539 343.076096 \nL 306.287848 343.214875 \nL 305.178572 343.568201 \nL 304.069296 343.920928 \nL 302.96002 344.273072 \nL 302.40795 344.44898 \nL 301.850743 344.621628 \nL 300.741467 344.966538 \nL 299.632191 345.310818 \nL 298.522915 345.654488 \nL 297.984784 345.821863 \nL 297.413638 345.994362 \nL 296.304362 346.330535 \nL 295.195086 346.666045 \nL 294.08581 347.00091 \nL 293.445597 347.194747 \nL 292.976533 347.332434 \nL 291.867257 347.659526 \nL 290.757981 347.985918 \nL 289.648705 348.311627 \nL 288.777268 348.567631 \nL 288.539428 348.635252 \nL 287.430152 348.952895 \nL 286.320876 349.269795 \nL 285.2116 349.585969 \nL 284.102323 349.90144 \nL 283.966128 349.940514 \nL 282.993047 350.210098 \nL 281.883771 350.517104 \nL 280.774495 350.823338 \nL 279.665218 351.128818 \nL 278.996818 351.313398 \nL 278.555942 351.430703 \nL 277.446666 351.727386 \nL 276.33739 352.02324 \nL 275.228113 352.318286 \nL 274.118837 352.612547 \nL 273.843018 352.686281 \nL 273.009561 352.900338 \nL 271.900285 353.185343 \nL 270.791008 353.469476 \nL 269.681732 353.75276 \nL 268.572456 354.035219 \nL 268.479395 354.059165 \nL 267.46318 354.309544 \nL 266.353903 354.58225 \nL 265.244627 354.854033 \nL 264.135351 355.124916 \nL 263.026074 355.394926 \nL 262.87511 355.432048 \nL 261.916798 355.656805 \nL 260.807522 355.916524 \nL 259.698246 356.175257 \nL 258.588969 356.433031 \nL 257.479693 356.68987 \nL 256.985499 356.804932 \nL 256.370417 356.940888 \nL 255.261141 357.186856 \nL 254.151864 357.431762 \nL 253.042588 357.675632 \nL 251.933312 357.918495 \nL 250.824036 358.160379 \nL 250.745071 358.177815 \nL 249.714759 358.392432 \nL 248.605483 358.622645 \nL 247.496207 358.851729 \nL 246.386931 359.079711 \nL 245.277654 359.306623 \nL 244.168378 359.532496 \nL 244.080162 359.550699 \nL 243.059102 359.74788 \nL 241.949826 359.961193 \nL 240.840549 360.173287 \nL 239.731273 360.384192 \nL 238.621997 360.593941 \nL 237.512721 360.802569 \nL 236.871643 360.923582 \nL 236.403444 361.005448 \nL 235.294168 361.20054 \nL 234.184892 361.394291 \nL 233.075616 361.586734 \nL 231.966339 361.777903 \nL 230.857063 361.967834 \nL 229.747787 362.156569 \nL 228.925486 362.296466 \nL 228.638511 362.340997 \nL 227.529234 362.514848 \nL 226.419958 362.687224 \nL 225.310682 362.858157 \nL 224.201406 363.027687 \nL 223.092129 363.195852 \nL 221.982853 363.362696 \nL 220.873577 363.528266 \nL 219.925016 363.669349 \nL 219.764301 363.690615 \nL 218.655024 363.839434 \nL 217.545748 363.986604 \nL 216.436472 364.132163 \nL 215.327196 364.27615 \nL 214.217919 364.418611 \nL 213.108643 364.559592 \nL 211.999367 364.699145 \nL 210.890091 364.837327 \nL 209.780814 364.974197 \nL 209.233526 365.042233 \nL 208.671538 365.101358 \nL 207.562262 365.2183 \nL 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299.143823 \nL 318.489887 298.323684 \nL 319.151495 297.77094 \nL 319.599163 297.39061 \nL 320.70844 296.453289 \nL 320.774311 296.398056 \nL 321.817716 295.50865 \nL 322.387742 295.025173 \nL 322.926992 294.560312 \nL 323.985686 293.652289 \nL 324.036268 293.608205 \nL 325.145545 292.648874 \nL 325.575055 292.279406 \nL 326.254821 291.685388 \nL 327.1507 290.906522 \nL 327.364097 290.718092 \nL 328.473373 289.745056 \nL 328.71593 289.533639 \nL 329.58265 288.766557 \nL 330.270682 288.160755 \nL 330.691926 287.784219 \nL 331.801202 286.797909 \nL 331.812576 286.787871 \nL 332.910479 285.804545 \nL 333.347686 285.414988 \nL 334.019755 284.807326 \nL 334.870531 284.042104 \nL 335.129031 283.806208 \nL 336.238307 282.800082 \nL 336.383546 282.669221 \nL 337.347584 281.788151 \nL 337.888541 281.296337 \nL 338.45686 280.772318 \nL 339.382377 279.923454 \nL 339.566136 279.752551 \nL 340.675412 278.727431 \nL 340.867991 278.55057 \nL 341.784689 277.697069 \nL 342.345487 277.177687 \nL 342.893965 276.66278 \nL 343.812767 275.804803 \nL 344.003241 275.624537 \nL 345.112517 274.581195 \nL 345.272226 274.43192 \nL 346.221794 273.532581 \nL 346.72445 273.059036 \nL 347.33107 272.480031 \nL 348.167269 271.686153 \nL 348.440346 271.423523 \nL 349.549622 270.362682 \nL 349.601637 270.313269 \nL 350.658899 269.296026 \nL 351.030501 268.940386 \nL 351.768175 268.225438 \nL 352.450664 267.567502 \nL 352.877451 267.150902 \nL 353.862393 266.194618 \nL 353.986727 266.072401 \nL 355.096004 264.98878 \nL 355.268019 264.821735 \nL 356.20528 263.900401 \nL 356.667092 263.448851 \nL 357.314556 262.808091 \nL 358.058312 262.075968 \nL 358.423832 261.71184 \nL 359.441904 260.703084 \nL 359.533109 260.611638 \nL 360.642385 259.506329 \nL 360.820164 259.330201 \nL 361.751661 258.396518 \nL 362.192176 257.957317 \nL 362.860937 257.282795 \nL 363.557047 256.584434 \nL 363.970214 256.165154 \nL 364.914966 255.21155 \nL 365.07949 255.04359 \nL 366.188766 253.917606 \nL 366.267007 253.838667 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229.126763 \nL 390.592844 228.115849 \nL 390.924706 227.75388 \nL 391.70212 226.897885 \nL 392.173999 226.380996 \nL 392.811396 225.676222 \nL 393.418774 225.008112 \nL 393.920672 224.450868 \nL 394.659117 223.635229 \nL 395.029949 223.22183 \nL 395.895106 222.262345 \nL 396.139225 221.989117 \nL 396.139225 220.889462 \nL 396.139225 219.516578 \nL 396.139225 218.143695 \nL 396.139225 216.770811 \nL 396.139225 215.397928 \nL 396.139225 214.025044 \nL 396.139225 212.652161 \nL 396.139225 211.279277 \nL 396.139225 209.906394 \nL 396.139225 208.53351 \nL 396.139225 207.160627 \nL 396.139225 205.787743 \nL 396.139225 204.414859 \nL 396.139225 203.041976 \nL 396.139225 201.669092 \nL 396.139225 200.296209 \nL 396.139225 198.923325 \nL 396.139225 197.550442 \nL 396.139225 196.177558 \nL 396.139225 194.804675 \nL 396.139225 193.431791 \nL 396.139225 192.058908 \nL 396.139225 190.686024 \nL 396.139225 189.313141 \nL 396.139225 187.940257 \nL 396.139225 186.567373 \nL 396.139225 185.19449 \nL 396.139225 183.821606 \nL 396.139225 182.448723 \nL 396.139225 181.075839 \nL 396.139225 179.702956 \nL 396.139225 178.330072 \nL 396.139225 176.957189 \nL 396.139225 175.584305 \nL 396.139225 174.211422 \nL 396.139225 172.838538 \nL 396.139225 171.465655 \nL 396.139225 170.092771 \nL 396.139225 168.719888 \nL 396.139225 167.347004 \nL 396.139225 165.97412 \nL 396.139225 164.601237 \nL 396.139225 163.228353 \nL 396.139225 161.85547 \nL 396.139225 160.482586 \nL 396.139225 159.109703 \nL 396.139225 157.736819 \nL 396.139225 156.363936 \nL 396.139225 154.991052 \nL 396.139225 153.618169 \nL 396.139225 152.245285 \nL 396.139225 150.872402 \nL 396.139225 149.499518 \nL 396.139225 148.126635 \nL 396.139225 146.753751 \nL 396.139225 145.380867 \nL 396.139225 144.007984 \nL 396.139225 142.6351 \nL 396.139225 141.262217 \nL 396.139225 139.889333 \nL 396.139225 138.51645 \nL 396.139225 137.143566 \nL 396.139225 135.770683 \nL 396.139225 134.397799 \nL 396.139225 133.024916 \nL 396.139225 131.652032 \nL 396.139225 130.279149 \nL 396.139225 128.906265 \nL 396.139225 127.533382 \nL 396.139225 126.160498 \nL 396.139225 124.787614 \nL 396.139225 123.414731 \nL 396.139225 122.041847 \nL 396.139225 120.668964 \nL 396.139225 119.29608 \nL 396.139225 117.923197 \nL 396.139225 116.550313 \nL 396.139225 115.17743 \nL 396.139225 113.804546 \nL 396.139225 112.431663 \nL 396.139225 111.058779 \nL 396.139225 109.685896 \nL 396.139225 108.313012 \nL 396.139225 106.940129 \nL 396.139225 105.567245 \nL 396.139225 104.194361 \nL 396.139225 102.821478 \nL 396.139225 101.448594 \nL 396.139225 100.075711 \nL 396.139225 98.702827 \nL 396.139225 97.329944 \nL 396.139225 96.837161 \nL 395.906217 97.329944 \nL 395.253236 98.702827 \nL 395.029949 99.16954 \nL 394.599429 100.075711 \nL 393.942958 101.448594 \nL 393.920672 101.494949 \nL 393.287384 102.821478 \nL 392.811396 103.811995 \nL 392.628942 104.194361 \nL 391.969971 105.567245 \nL 391.70212 106.121866 \nL 391.309729 106.940129 \nL 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139.889333 \nL 374.349473 141.262217 \nL 373.9537 142.038112 \nL 373.651484 142.6351 \nL 372.951439 144.007984 \nL 372.844424 144.216458 \nL 372.251232 145.380867 \nL 371.735147 146.386224 \nL 371.54792 146.753751 \nL 370.843616 148.126635 \nL 370.625871 148.54806 \nL 370.138026 149.499518 \nL 369.516595 150.702155 \nL 369.429301 150.872402 \nL 368.720521 152.245285 \nL 368.407319 152.847377 \nL 368.009455 153.618169 \nL 367.298042 154.985589 \nL 367.295222 154.991052 \nL 366.581737 156.363936 \nL 366.188766 157.114097 \nL 365.865095 157.736819 \nL 365.145923 159.109703 \nL 365.07949 159.235633 \nL 364.426819 160.482586 \nL 363.970214 161.347913 \nL 363.704497 161.85547 \nL 362.980155 163.228353 \nL 362.860937 163.452662 \nL 362.255295 164.601237 \nL 361.751661 165.548519 \nL 361.52718 165.97412 \nL 360.797394 167.347004 \nL 360.642385 167.636401 \nL 360.066659 168.719888 \nL 359.533109 169.715617 \nL 359.332632 170.092771 \nL 358.597108 171.465655 \nL 358.423832 171.786561 \nL 357.860372 172.838538 \nL 357.314556 173.848913 \nL 357.120307 174.211422 \nL 356.378728 175.584305 \nL 356.20528 175.902856 \nL 355.635858 176.957189 \nL 355.096004 177.948124 \nL 354.889617 178.330072 \nL 354.141646 179.702956 \nL 353.986727 179.985012 \nL 353.392497 181.075839 \nL 352.877451 182.012972 \nL 352.639933 182.448723 \nL 351.885206 183.821606 \nL 351.768175 184.032764 \nL 351.129623 185.19449 \nL 350.658899 186.043192 \nL 350.370579 186.567373 \nL 349.608704 187.940257 \nL 349.549622 188.045859 \nL 348.84652 189.313141 \nL 348.440346 190.038531 \nL 348.080825 190.686024 \nL 347.33107 192.023871 \nL 347.3116 192.058908 \nL 346.542415 193.431791 \nL 346.221794 193.99875 \nL 345.769885 194.804675 \nL 345.112517 195.965983 \nL 344.993771 196.177558 \nL 344.216472 197.550442 \nL 344.003241 197.923629 \nL 343.436907 198.923325 \nL 342.893965 199.872529 \nL 342.653699 200.296209 \nL 341.867783 201.669092 \nL 341.784689 201.812965 \nL 341.080965 203.041976 \nL 340.675412 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290.906522 \nL 281.883771 292.106357 \nL 281.747146 292.279406 \nL 280.774495 293.488218 \nL 280.64392 293.652289 \nL 279.665218 294.858605 \nL 279.53156 295.025173 \nL 278.555942 296.217467 \nL 278.409795 296.398056 \nL 277.446666 297.564756 \nL 277.278333 297.77094 \nL 276.33739 298.900429 \nL 276.136863 299.143823 \nL 275.228113 300.224445 \nL 274.985054 300.516707 \nL 274.118837 301.53677 \nL 273.822548 301.88959 \nL 273.009561 302.837374 \nL 272.648963 303.262474 \nL 271.900285 304.126233 \nL 271.463888 304.635357 \nL 270.791008 305.403328 \nL 270.266878 306.008241 \nL 269.681732 306.668648 \nL 269.057455 307.381124 \nL 268.572456 307.922191 \nL 267.835099 308.754008 \nL 267.46318 309.16396 \nL 266.59925 310.126892 \nL 266.353903 310.39397 \nL 265.349295 311.499775 \nL 265.244627 311.612246 \nL 264.135351 312.817969 \nL 264.085643 312.872659 \nL 263.026074 314.00999 \nL 262.80909 314.245542 \nL 261.916798 315.190096 \nL 261.51669 315.618426 \nL 260.807522 316.358344 \nL 260.207607 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343.439747 86.346876 \nL 342.980892 87.719759 \nL 342.893965 87.978444 \nL 342.52294 89.092643 \nL 342.063006 90.465526 \nL 341.784689 91.291438 \nL 341.602044 91.83841 \nL 341.141002 93.211293 \nL 340.677034 94.584177 \nL 340.675412 94.588948 \nL 340.214835 95.95706 \nL 339.74969 97.329944 \nL 339.566136 97.868576 \nL 339.284458 98.702827 \nL 338.818103 100.075711 \nL 338.45686 101.13253 \nL 338.349822 101.448594 \nL 337.882223 102.821478 \nL 337.411544 104.194361 \nL 337.347584 104.379866 \nL 336.942001 105.567245 \nL 336.47 106.940129 \nL 336.238307 107.609932 \nL 335.997383 108.313012 \nL 335.524024 109.685896 \nL 335.129031 110.824 \nL 335.048316 111.058779 \nL 334.573559 112.431663 \nL 334.09555 113.804546 \nL 334.019755 114.020954 \nL 333.61855 115.17743 \nL 333.139057 116.550313 \nL 332.910479 117.200641 \nL 332.658938 117.923197 \nL 332.177917 119.29608 \nL 331.801202 120.364011 \nL 331.694663 120.668964 \nL 331.212069 122.041847 \nL 330.726032 123.414731 \nL 330.691926 123.510489 \nL 330.241449 124.787614 \nL 329.753742 126.160498 \nL 329.58265 126.639022 \nL 329.26599 127.533382 \nL 328.776563 128.906265 \nL 328.473373 129.750927 \nL 328.285626 130.279149 \nL 327.794426 131.652032 \nL 327.364097 132.846091 \nL 327.300285 133.024916 \nL 326.807258 134.397799 \nL 326.310518 135.770683 \nL 326.254821 135.923632 \nL 325.814985 137.143566 \nL 325.316302 138.51645 \nL 325.145545 138.983367 \nL 324.817528 139.889333 \nL 324.316844 141.262217 \nL 324.036268 142.026068 \nL 323.814808 142.6351 \nL 323.312058 144.007984 \nL 322.926992 145.051622 \nL 322.80674 145.380867 \nL 322.30186 146.753751 \nL 321.817716 148.059918 \nL 321.793238 148.126635 \nL 321.286161 149.499518 \nL 320.774986 150.872402 \nL 320.70844 151.049899 \nL 320.264866 152.245285 \nL 319.75135 153.618169 \nL 319.599163 154.022116 \nL 319.23788 154.991052 \nL 318.721948 156.363936 \nL 318.489887 156.976811 \nL 318.205102 157.736819 \nL 317.686674 159.109703 \nL 317.380611 159.913881 \nL 317.166428 160.482586 \nL 316.64542 161.85547 \nL 316.271335 162.833223 \nL 316.121749 163.228353 \nL 315.598075 164.601237 \nL 315.162058 165.734735 \nL 315.07095 165.97412 \nL 314.54452 167.347004 \nL 314.052782 168.618318 \nL 314.013914 168.719888 \nL 313.484633 170.092771 \nL 312.950604 171.465655 \nL 312.943506 171.483768 \nL 312.418287 172.838538 \nL 311.881212 174.211422 \nL 311.83423 174.330602 \nL 311.345346 175.584305 \nL 310.805117 176.957189 \nL 310.724953 177.159313 \nL 310.26567 178.330072 \nL 309.722176 179.702956 \nL 309.615677 179.969815 \nL 309.179114 181.075839 \nL 308.632236 182.448723 \nL 308.506401 182.762026 \nL 308.085523 183.821606 \nL 307.535137 185.19449 \nL 307.397125 185.535867 \nL 306.984735 186.567373 \nL 306.430711 187.940257 \nL 306.287848 188.291262 \nL 305.876581 189.313141 \nL 305.318783 190.686024 \nL 305.178572 191.028141 \nL 304.760882 192.058908 \nL 304.199166 193.431791 \nL 304.069296 193.746436 \nL 303.637452 194.804675 \nL 303.071667 196.177558 \nL 302.96002 196.446086 \nL 302.506093 197.550442 \nL 301.936079 198.923325 \nL 301.850743 199.127036 \nL 301.366595 200.296209 \nL 300.792185 201.669092 \nL 300.741467 201.789235 \nL 300.218741 203.041976 \nL 299.639757 204.414859 \nL 299.632191 204.43264 \nL 299.062296 205.787743 \nL 298.522915 207.056487 \nL 298.479161 207.160627 \nL 297.897017 208.53351 \nL 297.413638 209.66119 \nL 297.309774 209.906394 \nL 296.722641 211.279277 \nL 296.304362 212.246843 \nL 296.131228 212.652161 \nL 295.538895 214.025044 \nL 295.195086 214.813428 \nL 294.943244 215.397928 \nL 294.345484 216.770811 \nL 294.08581 217.360933 \nL 293.745523 218.143695 \nL 293.142098 219.516578 \nL 292.976533 219.889359 \nL 292.53775 220.889462 \nL 291.928405 222.262345 \nL 291.867257 222.398715 \nL 291.319588 223.635229 \nL 290.757981 224.888092 \nL 290.704844 225.008112 \nL 290.090677 226.380996 \nL 289.648705 227.357187 \nL 289.471332 227.75388 \nL 288.850634 229.126763 \nL 288.539428 229.807089 \nL 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235.294168 266.714672 \nL 235.465927 266.194618 \nL 235.912382 264.821735 \nL 236.35177 263.448851 \nL 236.403444 263.284994 \nL 236.792483 262.075968 \nL 237.227488 260.703084 \nL 237.512721 259.789337 \nL 237.65892 259.330201 \nL 238.089806 257.957317 \nL 238.514317 256.584434 \nL 238.621997 256.231294 \nL 238.939089 255.21155 \nL 239.359929 253.838667 \nL 239.731273 252.609901 \nL 239.775677 252.465783 \nL 240.193058 251.0929 \nL 240.60464 249.720016 \nL 240.840549 248.922475 \nL 241.014008 248.347133 \nL 241.422497 246.974249 \nL 241.825522 245.601365 \nL 241.949826 245.172497 \nL 242.228632 244.228482 \nL 242.628888 242.855598 \nL 243.023981 241.482715 \nL 243.059102 241.359179 \nL 243.420938 240.109831 \nL 243.813551 238.736948 \nL 244.168378 237.480611 \nL 244.201902 237.364064 \nL 244.592171 235.991181 \nL 244.977669 234.618297 \nL 245.277654 233.537069 \nL 245.360048 233.245414 \nL 245.743444 231.87253 \nL 246.122306 230.499647 \nL 246.386931 229.529576 \nL 246.498789 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256.999517 187.940257 \nL 257.324722 186.567373 \nL 257.479693 185.907499 \nL 257.649787 185.19449 \nL 257.974502 183.821606 \nL 258.296433 182.448723 \nL 258.588969 181.190541 \nL 258.616055 181.075839 \nL 258.937568 179.702956 \nL 259.256386 178.330072 \nL 259.572559 176.957189 \nL 259.698246 176.406971 \nL 259.889063 175.584305 \nL 260.204918 174.211422 \nL 260.518207 172.838538 \nL 260.807522 171.560459 \nL 260.829309 171.465655 \nL 261.142341 170.092771 \nL 261.452883 168.719888 \nL 261.760981 167.347004 \nL 261.916798 166.647274 \nL 262.068951 165.97412 \nL 262.376873 164.601237 \nL 262.682419 163.228353 \nL 262.985631 161.85547 \nL 263.026074 161.67097 \nL 263.290445 160.482586 \nL 263.593555 159.109703 \nL 263.894394 157.736819 \nL 264.135351 156.62901 \nL 264.193853 156.363936 \nL 264.494635 154.991052 \nL 264.793206 153.618169 \nL 265.089603 152.245285 \nL 265.244627 151.522038 \nL 265.385892 150.872402 \nL 265.682294 149.499518 \nL 265.976577 148.126635 \nL 266.268776 146.753751 \nL 266.353903 146.350973 \nL 266.561873 145.380867 \nL 266.854133 144.007984 \nL 267.144359 142.6351 \nL 267.432582 141.262217 \nL 267.46318 141.115479 \nL 267.722468 139.889333 \nL 268.010803 138.51645 \nL 268.297185 137.143566 \nL 268.572456 135.815018 \nL 268.581771 135.770683 \nL 268.868294 134.397799 \nL 269.152909 133.024916 \nL 269.435643 131.652032 \nL 269.681732 130.449209 \nL 269.717007 130.279149 \nL 269.999926 128.906265 \nL 270.281007 127.533382 \nL 270.560276 126.160498 \nL 270.791008 125.018925 \nL 270.8384 124.787614 \nL 271.117894 123.414731 \nL 271.395614 122.041847 \nL 271.671588 120.668964 \nL 271.900285 119.524123 \nL 271.946455 119.29608 \nL 272.222689 117.923197 \nL 272.497211 116.550313 \nL 272.770047 115.17743 \nL 273.009561 113.964822 \nL 273.041642 113.804546 \nL 273.31477 112.431663 \nL 273.586245 111.058779 \nL 273.856089 109.685896 \nL 274.118837 108.341101 \nL 274.124398 108.313012 \nL 274.394564 106.940129 \nL 274.663131 105.567245 \nL 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283.131114 60.262088 \nL 283.378041 58.889205 \nL 283.623749 57.516321 \nL 283.868253 56.143438 \nL 284.102323 54.822701 \nL 284.111676 54.770554 \nL 284.356689 53.397671 \nL 284.600514 52.024787 \nL 284.843165 50.651904 \nL 285.084654 49.27902 \nL 285.2116 48.553844 \nL 285.32633 47.906137 \nL 285.568343 46.533253 \nL 285.80921 45.160369 \nL 286.048945 43.787486 \nL 286.287559 42.414602 \nL 286.320876 42.222 \nL 286.527439 41.041719 \nL 286.76659 39.668835 \nL 287.004634 38.295952 \nL 287.241585 36.923068 \nL 287.430152 35.82548 \nL 287.477998 35.550185 \nL 287.715494 34.177301 \nL 287.95191 32.804418 \nL 288.187258 31.431534 \nL 288.421548 30.058651 \nL 288.539428 29.364757 \nL 288.656105 28.685767 \nL 288.89095 27.312884 \nL 289.124752 25.94 \nL 288.539428 25.94 \nL 287.430152 25.94 \nL 286.320876 25.94 \nL 285.2116 25.94 \nL 284.102323 25.94 \nL 282.993047 25.94 \nL 281.883771 25.94 \nL 280.774495 25.94 \nL 279.665218 25.94 \nL 278.555942 25.94 \nL 277.446666 25.94 \nL 276.33739 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            "text/plain": [
              "\u003cFigure size 690x600 with 2 Axes\u003e"
            ]
          },
          "metadata": {
            "tags": []
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "W_var = jnp.arange(0, 3, 0.01)\n",
        "b_var = jnp.arange(0., 0.25, 0.001)\n",
        "\n",
        "plt.contourf(W_var, b_var, c_star(W_var, b_var))\n",
        "plt.colorbar()\n",
        "plt.title('$C^*$ as a function of weight and bias variance', fontsize=14)\n",
        "\n",
        "format_plot('$\\\\sigma_w^2$', '$\\\\sigma_b^2$')\n",
        "finalize_plot((1.15, 1))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "colab_type": "text",
        "id": "-YLDRMt5F6mu"
      },
      "source": [
        "We can, of course, threshold on $c^*$ to get a cleaner definition of the phase diagram."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 0,
      "metadata": {
        "colab": {
          "height": 582
        },
        "colab_type": "code",
        "executionInfo": {
          "elapsed": 668,
          "status": "ok",
          "timestamp": 1579989871731,
          "user": {
            "displayName": "",
            "photoUrl": "",
            "userId": ""
          },
          "user_tz": 480
        },
        "id": "Cr7MFuCs1Y-3",
        "outputId": "4a5d5d02-9234-410b-ebcb-c3effef58ea6"
      },
      "outputs": [
        {
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254.955469 \nL 399.092625 253.510984 \nL 399.092625 252.066499 \nL 399.092625 250.622013 \nL 399.092625 249.177528 \nL 399.092625 247.733043 \nL 399.092625 246.288557 \nL 399.092625 244.844072 \nL 399.092625 243.399587 \nL 399.092625 241.955101 \nL 399.092625 240.510616 \nL 399.092625 239.06613 \nL 399.092625 237.621645 \nL 399.092625 236.17716 \nL 399.092625 234.732674 \nL 399.092625 233.288189 \nL 399.092625 231.843704 \nL 399.092625 230.399218 \nL 399.092625 228.954733 \nL 399.092625 227.510248 \nL 399.092625 226.065762 \nL 399.092625 224.621277 \nL 399.092625 223.176792 \nL 399.092625 221.732306 \nL 399.092625 220.287821 \nL 399.092625 218.843336 \nL 399.092625 217.39885 \nL 399.092625 215.954365 \nL 399.092625 214.509879 \nL 399.092625 213.065394 \nL 399.092625 211.620909 \nL 399.092625 210.176423 \nL 399.092625 208.731938 \nL 399.092625 207.287453 \nL 399.092625 205.842967 \nL 399.092625 204.398482 \nL 399.092625 202.953997 \nL 399.092625 201.509511 \nL 399.092625 200.065026 \nL 399.092625 198.620541 \nL 399.092625 197.176055 \nL 399.092625 195.73157 \nL 399.092625 194.287084 \nL 399.092625 192.842599 \nL 399.092625 191.398114 \nL 399.092625 189.953628 \nL 399.092625 188.509143 \nL 399.092625 187.064658 \nL 399.092625 185.620172 \nL 399.092625 184.175687 \nL 399.092625 182.731202 \nL 399.092625 181.286716 \nL 399.092625 179.842231 \nL 399.092625 178.397746 \nL 399.092625 176.95326 \nL 399.092625 175.508775 \nL 399.092625 174.064289 \nL 399.092625 172.619804 \nL 399.092625 171.175319 \nL 399.092625 169.730833 \nL 399.092625 168.286348 \nL 399.092625 166.841863 \nL 399.092625 165.397377 \nL 399.092625 163.952892 \nL 399.092625 162.508407 \nL 399.092625 161.063921 \nL 399.092625 159.619436 \nL 399.092625 158.174951 \nL 399.092625 156.730465 \nL 399.092625 155.28598 \nL 399.092625 153.841495 \nL 399.092625 152.397009 \nL 399.092625 150.952524 \nL 399.092625 149.508038 \nL 399.092625 148.063553 \nL 399.092625 146.619068 \nL 399.092625 145.174582 \nL 399.092625 143.730097 \nL 399.092625 142.285612 \nL 399.092625 140.841126 \nL 399.092625 139.396641 \nL 399.092625 137.952156 \nL 399.092625 136.50767 \nL 399.092625 135.063185 \nL 399.092625 133.6187 \nL 399.092625 132.174214 \nL 399.092625 130.729729 \nL 399.092625 129.285243 \nL 399.092625 127.840758 \nL 399.092625 126.396273 \nL 399.092625 124.951787 \nL 399.092625 123.507302 \nL 399.092625 122.062817 \nL 399.092625 120.618331 \nL 399.092625 119.173846 \nL 399.092625 117.729361 \nL 399.092625 116.284875 \nL 399.092625 114.84039 \nL 399.092625 113.395905 \nL 399.092625 111.951419 \nL 399.092625 110.506934 \nL 399.092625 109.062448 \nL 399.092625 107.617963 \nL 399.092625 106.173478 \nL 399.092625 104.728992 \nL 399.092625 103.284507 \nL 399.092625 101.840022 \nL 399.092625 100.395536 \nL 399.092625 98.951051 \nL 399.092625 97.506566 \nL 399.092625 96.06208 \nL 399.092625 94.617595 \nL 399.092625 93.17311 \nL 399.092625 91.728624 \nL 399.092625 90.284139 \nL 399.092625 88.839654 \nL 399.092625 87.395168 \nL 399.092625 85.950683 \nL 399.092625 84.506197 \nL 399.092625 83.061712 \nL 399.092625 81.617227 \nL 399.092625 80.172741 \nL 399.092625 78.728256 \nL 399.092625 77.283771 \nL 399.092625 75.839285 \nL 399.092625 74.3948 \nL 399.092625 72.950315 \nL 399.092625 71.505829 \nL 399.092625 70.061344 \nL 399.092625 68.616859 \nL 399.092625 67.172373 \nL 399.092625 65.727888 \nL 399.092625 64.283402 \nL 399.092625 62.838917 \nL 399.092625 61.394432 \nL 399.092625 59.949946 \nL 399.092625 58.505461 \nL 399.092625 57.060976 \nL 399.092625 55.61649 \nL 399.092625 54.172005 \nL 399.092625 52.72752 \nL 399.092625 51.283034 \nL 399.092625 49.838549 \nL 399.092625 48.394064 \nL 399.092625 46.949578 \nL 399.092625 45.505093 \nL 399.092625 44.060607 \nL 399.092625 42.616122 \nL 399.092625 41.171637 \nL 399.092625 39.727151 \nL 399.092625 38.282666 \nL 399.092625 36.838181 \nL 399.092625 35.393695 \nL 399.092625 33.94921 \nL 399.092625 32.504725 \nL 399.092625 31.060239 \nL 399.092625 29.615754 \nL 399.092625 28.171269 \nL 399.092625 26.726783 \nL 399.092625 25.282298 \nL 399.092625 23.837812 \nL 397.973471 23.837812 \nL 396.854317 23.837812 \nL 395.735163 23.837812 \nL 394.61601 23.837812 \nL 393.496856 23.837812 \nL 392.377702 23.837812 \nL 391.258548 23.837812 \nL 390.139394 23.837812 \nL 389.02024 23.837812 \nL 387.901087 23.837812 \nL 386.781933 23.837812 \nL 385.662779 23.837812 \nL 384.543625 23.837812 \nL 383.424471 23.837812 \nL 382.305317 23.837812 \nL 381.186163 23.837812 \nL 380.06701 23.837812 \nL 378.947856 23.837812 \nL 377.828702 23.837812 \nL 376.709548 23.837812 \nL 375.590394 23.837812 \nL 374.47124 23.837812 \nL 373.352087 23.837812 \nL 372.232933 23.837812 \nL 371.113779 23.837812 \nL 369.994625 23.837812 \nL 368.875471 23.837812 \nL 367.756317 23.837812 \nL 366.637163 23.837812 \nL 365.51801 23.837812 \nL 364.398856 23.837812 \nL 363.279702 23.837812 \nL 362.160548 23.837812 \nL 361.041394 23.837812 \nL 359.92224 23.837812 \nL 358.803087 23.837812 \nL 357.683933 23.837812 \nL 356.564779 23.837812 \nL 355.445625 23.837812 \nL 354.326471 23.837812 \nL 353.207317 23.837812 \nL 352.088163 23.837812 \nL 350.96901 23.837812 \nL 349.849856 23.837812 \nL 348.730702 23.837812 \nL 347.611548 23.837812 \nL 346.492394 23.837812 \nL 345.37324 23.837812 \nL 344.254087 23.837812 \nL 343.134933 23.837812 \nL 342.015779 23.837812 \nL 340.896625 23.837812 \nL 339.777471 23.837812 \nL 338.658317 23.837812 \nL 337.539163 23.837812 \nL 336.42001 23.837812 \nL 335.300856 23.837812 \nL 334.181702 23.837812 \nL 333.062548 23.837812 \nL 331.943394 23.837812 \nL 330.82424 23.837812 \nL 329.705087 23.837812 \nL 328.585933 23.837812 \nL 327.466779 23.837812 \nL 326.347625 23.837812 \nL 325.228471 23.837812 \nL 324.109317 23.837812 \nL 322.990163 23.837812 \nL 321.87101 23.837812 \nL 320.751856 23.837812 \nL 319.632702 23.837812 \nL 318.513548 23.837812 \nL 317.394394 23.837812 \nL 316.27524 23.837812 \nL 315.156087 23.837812 \nL 314.036933 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267.032471 79.81162 \nL 266.752683 80.172741 \nL 266.752683 81.617227 \nL 266.752683 83.061712 \nL 266.752683 84.506197 \nL 266.752683 85.950683 \nL 265.913317 87.034047 \nL 265.633529 87.395168 \nL 265.633529 88.839654 \nL 265.633529 90.284139 \nL 265.633529 91.728624 \nL 265.633529 93.17311 \nL 264.794163 94.256474 \nL 264.514375 94.617595 \nL 264.514375 96.06208 \nL 264.514375 97.506566 \nL 264.514375 98.951051 \nL 264.514375 100.395536 \nL 263.67501 101.4789 \nL 263.395221 101.840022 \nL 263.395221 103.284507 \nL 263.395221 104.728992 \nL 263.395221 106.173478 \nL 263.395221 107.617963 \nL 262.555856 108.701327 \nL 262.276067 109.062448 \nL 262.276067 110.506934 \nL 262.276067 111.951419 \nL 262.276067 113.395905 \nL 262.276067 114.84039 \nL 261.436702 115.923754 \nL 261.156913 116.284875 \nL 261.156913 117.729361 \nL 261.156913 119.173846 \nL 261.156913 120.618331 \nL 261.156913 122.062817 \nL 260.317548 123.146181 \nL 260.03776 123.507302 \nL 260.03776 124.951787 \nL 260.03776 126.396273 \nL 260.03776 127.840758 \nL 260.03776 129.285243 \nL 259.198394 130.368607 \nL 258.918606 130.729729 \nL 258.918606 132.174214 \nL 258.918606 133.6187 \nL 258.918606 135.063185 \nL 258.07924 136.146549 \nL 257.799452 136.50767 \nL 257.799452 137.952156 \nL 257.799452 139.396641 \nL 257.799452 140.841126 \nL 257.799452 142.285612 \nL 256.960087 143.368976 \nL 256.680298 143.730097 \nL 256.680298 145.174582 \nL 256.680298 146.619068 \nL 256.680298 148.063553 \nL 255.840933 149.146917 \nL 255.561144 149.508038 \nL 255.561144 150.952524 \nL 255.561144 152.397009 \nL 255.561144 153.841495 \nL 255.561144 155.28598 \nL 254.721779 156.369344 \nL 254.44199 156.730465 \nL 254.44199 158.174951 \nL 254.44199 159.619436 \nL 254.44199 161.063921 \nL 253.602625 162.147285 \nL 253.322837 162.508407 \nL 253.322837 163.952892 \nL 253.322837 165.397377 \nL 253.322837 166.841863 \nL 253.322837 168.286348 \nL 252.483471 169.369712 \nL 252.203683 169.730833 \nL 252.203683 171.175319 \nL 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            "text/plain": [
              "\u003cFigure size 600x600 with 1 Axes\u003e"
            ]
          },
          "metadata": {
            "tags": []
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "plt.contourf(W_var, b_var, c_star(W_var, b_var) \u003e 0.999, \n",
        "             levels=3, \n",
        "             colors=[[1.0, 0.89, 0.811], [0.85, 0.85, 1]])\n",
        "plt.title('Phase diagram in terms of weight and bias variance', fontsize=14)\n",
        "\n",
        "format_plot('$\\\\sigma_w^2$', '$\\\\sigma_b^2$')\n",
        "finalize_plot((1, 1))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "colab_type": "text",
        "id": "vV8gMoESLbWo"
      },
      "source": [
        "As described above, the boundary between the two phases should be defined by $\\chi_1(\\sigma_w^2, \\sigma_b^2) = 1$ where $\\chi_1$ is given by the derivative of the $\\mathcal C$-map."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 0,
      "metadata": {
        "colab": {
          "height": 580
        },
        "colab_type": "code",
        "executionInfo": {
          "elapsed": 2088,
          "status": "ok",
          "timestamp": 1579989876633,
          "user": {
            "displayName": "",
            "photoUrl": "",
            "userId": ""
          },
          "user_tz": 480
        },
        "id": "u4e4j-wvO315",
        "outputId": "03e99c83-14b1-4006-cf66-af6d1e051e69"
      },
      "outputs": [
        {
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            "text/plain": [
              "\u003cFigure size 690x600 with 2 Axes\u003e"
            ]
          },
          "metadata": {
            "tags": []
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "plt.contourf(W_var, b_var, chi_1(W_var, b_var))\n",
        "plt.colorbar()\n",
        "plt.title(r'$\\chi^1$ as a function of weight and bias variance', fontsize=14)\n",
        "\n",
        "format_plot('$\\\\sigma_w^2$', '$\\\\sigma_b^2$')\n",
        "finalize_plot((1.15, 1))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "colab_type": "text",
        "id": "_7I9hm_4PBNl"
      },
      "source": [
        "We can see that the boundary where $\\chi_1$ crosses 1 corresponds to the phase boundary we observe above."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 0,
      "metadata": {
        "colab": {
          "height": 582
        },
        "colab_type": "code",
        "executionInfo": {
          "elapsed": 666,
          "status": "ok",
          "timestamp": 1579934487006,
          "user": {
            "displayName": "",
            "photoUrl": "",
            "userId": ""
          },
          "user_tz": 480
        },
        "id": "-4SgMiPXPNhl",
        "outputId": "890b7531-7ed0-452e-f750-3b20f043c566"
      },
      "outputs": [
        {
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369.069812 \nL 316.27524 369.069812 \nL 317.394394 369.069812 \nL 318.513548 369.069812 \nL 319.632702 369.069812 \nL 320.751856 369.069812 \nL 321.87101 369.069812 \nL 322.990163 369.069812 \nL 324.109317 369.069812 \nL 325.228471 369.069812 \nL 326.347625 369.069812 \nL 327.466779 369.069812 \nL 328.585933 369.069812 \nL 329.705087 369.069812 \nL 330.82424 369.069812 \nL 331.943394 369.069812 \nL 333.062548 369.069812 \nL 334.181702 369.069812 \nL 335.300856 369.069812 \nL 336.42001 369.069812 \nL 337.539163 369.069812 \nL 338.658317 369.069812 \nL 339.777471 369.069812 \nL 340.896625 369.069812 \nL 342.015779 369.069812 \nL 343.134933 369.069812 \nL 344.254087 369.069812 \nL 345.37324 369.069812 \nL 346.492394 369.069812 \nL 347.611548 369.069812 \nL 348.730702 369.069812 \nL 349.849856 369.069812 \nL 350.96901 369.069812 \nL 352.088163 369.069812 \nL 353.207317 369.069812 \nL 354.326471 369.069812 \nL 355.445625 369.069812 \nL 356.564779 369.069812 \nL 357.683933 369.069812 \nL 358.803087 369.069812 \nL 359.92224 369.069812 \nL 361.041394 369.069812 \nL 362.160548 369.069812 \nL 363.279702 369.069812 \nL 364.398856 369.069812 \nL 365.51801 369.069812 \nL 366.637163 369.069812 \nL 367.756317 369.069812 \nL 368.875471 369.069812 \nL 369.994625 369.069812 \nL 371.113779 369.069812 \nL 372.232933 369.069812 \nL 373.352087 369.069812 \nL 374.47124 369.069812 \nL 375.590394 369.069812 \nL 376.709548 369.069812 \nL 377.828702 369.069812 \nL 378.947856 369.069812 \nL 380.06701 369.069812 \nL 381.186163 369.069812 \nL 382.305317 369.069812 \nL 383.424471 369.069812 \nL 384.543625 369.069812 \nL 385.662779 369.069812 \nL 386.781933 369.069812 \nL 387.901087 369.069812 \nL 389.02024 369.069812 \nL 390.139394 369.069812 \nL 391.258548 369.069812 \nL 392.377702 369.069812 \nL 393.496856 369.069812 \nL 394.61601 369.069812 \nL 395.735163 369.069812 \nL 396.854317 369.069812 \nL 397.973471 369.069812 \nL 399.092625 369.069812 \nL 399.092625 367.625327 \nL 399.092625 366.180842 \nL 399.092625 364.736356 \nL 399.092625 363.291871 \nL 399.092625 361.847386 \nL 399.092625 360.4029 \nL 399.092625 358.958415 \nL 399.092625 357.51393 \nL 399.092625 356.069444 \nL 399.092625 354.624959 \nL 399.092625 353.180474 \nL 399.092625 351.735988 \nL 399.092625 350.291503 \nL 399.092625 348.847018 \nL 399.092625 347.402532 \nL 399.092625 345.958047 \nL 399.092625 344.513561 \nL 399.092625 343.069076 \nL 399.092625 341.624591 \nL 399.092625 340.180105 \nL 399.092625 338.73562 \nL 399.092625 337.291135 \nL 399.092625 335.846649 \nL 399.092625 334.402164 \nL 399.092625 332.957679 \nL 399.092625 331.513193 \nL 399.092625 330.068708 \nL 399.092625 328.624223 \nL 399.092625 327.179737 \nL 399.092625 325.735252 \nL 399.092625 324.290766 \nL 399.092625 322.846281 \nL 399.092625 321.401796 \nL 399.092625 319.95731 \nL 399.092625 318.512825 \nL 399.092625 317.06834 \nL 399.092625 315.623854 \nL 399.092625 314.179369 \nL 399.092625 312.734884 \nL 399.092625 311.290398 \nL 399.092625 309.845913 \nL 399.092625 308.401428 \nL 399.092625 306.956942 \nL 399.092625 305.512457 \nL 399.092625 304.067971 \nL 399.092625 302.623486 \nL 399.092625 301.179001 \nL 399.092625 299.734515 \nL 399.092625 298.29003 \nL 399.092625 296.845545 \nL 399.092625 295.401059 \nL 399.092625 293.956574 \nL 399.092625 292.512089 \nL 399.092625 291.067603 \nL 399.092625 289.623118 \nL 399.092625 288.178633 \nL 399.092625 286.734147 \nL 399.092625 285.289662 \nL 399.092625 283.845177 \nL 399.092625 282.400691 \nL 399.092625 280.956206 \nL 399.092625 279.51172 \nL 399.092625 278.067235 \nL 399.092625 276.62275 \nL 399.092625 275.178264 \nL 399.092625 273.733779 \nL 399.092625 272.289294 \nL 399.092625 270.844808 \nL 399.092625 269.400323 \nL 399.092625 267.955838 \nL 399.092625 266.511352 \nL 399.092625 265.066867 \nL 399.092625 263.622382 \nL 399.092625 262.177896 \nL 399.092625 260.733411 \nL 399.092625 259.288925 \nL 399.092625 257.84444 \nL 399.092625 256.399955 \nL 399.092625 254.955469 \nL 399.092625 253.510984 \nL 399.092625 252.066499 \nL 399.092625 250.622013 \nL 399.092625 249.177528 \nL 399.092625 247.733043 \nL 399.092625 246.288557 \nL 399.092625 244.844072 \nL 399.092625 243.399587 \nL 399.092625 241.955101 \nL 399.092625 240.510616 \nL 399.092625 239.06613 \nL 399.092625 237.621645 \nL 399.092625 236.17716 \nL 399.092625 234.732674 \nL 399.092625 233.288189 \nL 399.092625 231.843704 \nL 399.092625 230.399218 \nL 399.092625 228.954733 \nL 399.092625 227.510248 \nL 399.092625 226.065762 \nL 399.092625 224.621277 \nL 399.092625 223.176792 \nL 399.092625 221.732306 \nL 399.092625 220.287821 \nL 399.092625 218.843336 \nL 399.092625 217.39885 \nL 399.092625 215.954365 \nL 399.092625 214.509879 \nL 399.092625 213.065394 \nL 399.092625 211.620909 \nL 399.092625 210.176423 \nL 399.092625 208.731938 \nL 399.092625 207.287453 \nL 399.092625 205.842967 \nL 399.092625 204.398482 \nL 399.092625 202.953997 \nL 399.092625 201.509511 \nL 399.092625 200.065026 \nL 399.092625 198.620541 \nL 399.092625 197.176055 \nL 399.092625 195.73157 \nL 399.092625 194.287084 \nL 399.092625 192.842599 \nL 399.092625 191.398114 \nL 399.092625 189.953628 \nL 399.092625 188.509143 \nL 399.092625 187.064658 \nL 399.092625 185.620172 \nL 399.092625 184.175687 \nL 399.092625 182.731202 \nL 399.092625 181.286716 \nL 399.092625 179.842231 \nL 399.092625 178.397746 \nL 399.092625 176.95326 \nL 399.092625 175.508775 \nL 399.092625 174.064289 \nL 399.092625 172.619804 \nL 399.092625 171.175319 \nL 399.092625 169.730833 \nL 399.092625 168.286348 \nL 399.092625 166.841863 \nL 399.092625 165.397377 \nL 399.092625 163.952892 \nL 399.092625 162.508407 \nL 399.092625 161.063921 \nL 399.092625 159.619436 \nL 399.092625 158.174951 \nL 399.092625 156.730465 \nL 399.092625 155.28598 \nL 399.092625 153.841495 \nL 399.092625 152.397009 \nL 399.092625 150.952524 \nL 399.092625 149.508038 \nL 399.092625 148.063553 \nL 399.092625 146.619068 \nL 399.092625 145.174582 \nL 399.092625 143.730097 \nL 399.092625 142.285612 \nL 399.092625 140.841126 \nL 399.092625 139.396641 \nL 399.092625 137.952156 \nL 399.092625 136.50767 \nL 399.092625 135.063185 \nL 399.092625 133.6187 \nL 399.092625 132.174214 \nL 399.092625 130.729729 \nL 399.092625 129.285243 \nL 399.092625 127.840758 \nL 399.092625 126.396273 \nL 399.092625 124.951787 \nL 399.092625 123.507302 \nL 399.092625 122.062817 \nL 399.092625 120.618331 \nL 399.092625 119.173846 \nL 399.092625 117.729361 \nL 399.092625 116.284875 \nL 399.092625 114.84039 \nL 399.092625 113.395905 \nL 399.092625 111.951419 \nL 399.092625 110.506934 \nL 399.092625 109.062448 \nL 399.092625 107.617963 \nL 399.092625 106.173478 \nL 399.092625 104.728992 \nL 399.092625 103.284507 \nL 399.092625 101.840022 \nL 399.092625 100.395536 \nL 399.092625 98.951051 \nL 399.092625 97.506566 \nL 399.092625 96.06208 \nL 399.092625 94.617595 \nL 399.092625 93.17311 \nL 399.092625 91.728624 \nL 399.092625 90.284139 \nL 399.092625 88.839654 \nL 399.092625 87.395168 \nL 399.092625 85.950683 \nL 399.092625 84.506197 \nL 399.092625 83.061712 \nL 399.092625 81.617227 \nL 399.092625 80.172741 \nL 399.092625 78.728256 \nL 399.092625 77.283771 \nL 399.092625 75.839285 \nL 399.092625 74.3948 \nL 399.092625 72.950315 \nL 399.092625 71.505829 \nL 399.092625 70.061344 \nL 399.092625 68.616859 \nL 399.092625 67.172373 \nL 399.092625 65.727888 \nL 399.092625 64.283402 \nL 399.092625 62.838917 \nL 399.092625 61.394432 \nL 399.092625 59.949946 \nL 399.092625 58.505461 \nL 399.092625 57.060976 \nL 399.092625 55.61649 \nL 399.092625 54.172005 \nL 399.092625 52.72752 \nL 399.092625 51.283034 \nL 399.092625 49.838549 \nL 399.092625 48.394064 \nL 399.092625 46.949578 \nL 399.092625 45.505093 \nL 399.092625 44.060607 \nL 399.092625 42.616122 \nL 399.092625 41.171637 \nL 399.092625 39.727151 \nL 399.092625 38.282666 \nL 399.092625 36.838181 \nL 399.092625 35.393695 \nL 399.092625 33.94921 \nL 399.092625 32.504725 \nL 399.092625 31.060239 \nL 399.092625 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244.929183 207.287453 \nL 244.929183 205.842967 \nL 244.929183 204.398482 \nL 245.768548 203.315118 \nL 246.048337 202.953997 \nL 246.048337 201.509511 \nL 246.048337 200.065026 \nL 246.048337 198.620541 \nL 246.887702 197.537177 \nL 247.16749 197.176055 \nL 247.16749 195.73157 \nL 247.16749 194.287084 \nL 247.16749 192.842599 \nL 248.006856 191.759235 \nL 248.286644 191.398114 \nL 248.286644 189.953628 \nL 248.286644 188.509143 \nL 248.286644 187.064658 \nL 249.12601 185.981294 \nL 249.405798 185.620172 \nL 249.405798 184.175687 \nL 249.405798 182.731202 \nL 249.405798 181.286716 \nL 250.245163 180.203352 \nL 250.524952 179.842231 \nL 250.524952 178.397746 \nL 250.524952 176.95326 \nL 250.524952 175.508775 \nL 251.364317 174.425411 \nL 251.644106 174.064289 \nL 251.644106 172.619804 \nL 251.644106 171.175319 \nL 251.644106 169.730833 \nL 252.483471 168.647469 \nL 252.76326 168.286348 \nL 252.76326 166.841863 \nL 252.76326 165.397377 \nL 252.76326 163.952892 \nL 252.76326 162.508407 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116.284875 \nL 261.436702 115.201511 \nL 261.71649 114.84039 \nL 261.71649 113.395905 \nL 261.71649 111.951419 \nL 261.71649 110.506934 \nL 261.71649 109.062448 \nL 262.555856 107.979084 \nL 262.835644 107.617963 \nL 262.835644 106.173478 \nL 262.835644 104.728992 \nL 262.835644 103.284507 \nL 262.835644 101.840022 \nL 263.67501 100.756658 \nL 263.954798 100.395536 \nL 263.954798 98.951051 \nL 263.954798 97.506566 \nL 263.954798 96.06208 \nL 263.954798 94.617595 \nL 264.794163 93.534231 \nL 265.073952 93.17311 \nL 265.073952 91.728624 \nL 265.073952 90.284139 \nL 265.073952 88.839654 \nL 265.073952 87.395168 \nL 265.913317 86.311804 \nL 266.193106 85.950683 \nL 266.193106 84.506197 \nL 266.193106 83.061712 \nL 266.193106 81.617227 \nL 266.193106 80.172741 \nL 267.032471 79.089377 \nL 267.31226 78.728256 \nL 267.31226 77.283771 \nL 267.31226 75.839285 \nL 267.31226 74.3948 \nL 267.31226 72.950315 \nL 268.151625 71.866951 \nL 268.431413 71.505829 \nL 268.431413 70.061344 \nL 268.431413 68.616859 \nL 268.431413 67.172373 \nL 268.431413 65.727888 \nL 268.431413 64.283402 \nL 269.270779 63.200038 \nL 269.550567 62.838917 \nL 269.550567 61.394432 \nL 269.550567 59.949946 \nL 269.550567 58.505461 \nL 269.550567 57.060976 \nL 270.389933 55.977612 \nL 270.669721 55.61649 \nL 270.669721 54.172005 \nL 270.669721 52.72752 \nL 270.669721 51.283034 \nL 270.669721 49.838549 \nL 270.669721 48.394064 \nL 271.509087 47.3107 \nL 271.788875 46.949578 \nL 271.788875 45.505093 \nL 271.788875 44.060607 \nL 271.788875 42.616122 \nL 271.788875 41.171637 \nL 272.62824 40.088273 \nL 272.908029 39.727151 \nL 272.908029 38.282666 \nL 272.908029 36.838181 \nL 272.908029 35.393695 \nL 272.908029 33.94921 \nL 272.908029 32.504725 \nL 273.747394 31.421361 \nL 274.027183 31.060239 \nL 274.027183 29.615754 \nL 274.027183 28.171269 \nL 274.027183 26.726783 \nL 274.027183 25.282298 \nL 274.866548 24.198934 \nL 275.146337 23.837812 \nL 274.866548 23.837812 \nL 273.747394 23.837812 \nL 272.62824 23.837812 \nL 271.509087 23.837812 \nL 270.389933 23.837812 \nL 269.270779 23.837812 \nL 268.151625 23.837812 \nL 267.032471 23.837812 \nL 265.913317 23.837812 \nL 264.794163 23.837812 \nL 263.67501 23.837812 \nL 262.555856 23.837812 \nL 261.436702 23.837812 \nL 260.317548 23.837812 \nL 259.198394 23.837812 \nL 258.07924 23.837812 \nL 256.960087 23.837812 \nL 255.840933 23.837812 \nL 254.721779 23.837812 \nL 253.602625 23.837812 \nL 252.483471 23.837812 \nL 251.364317 23.837812 \nL 250.245163 23.837812 \nL 249.12601 23.837812 \nL 248.006856 23.837812 \nL 246.887702 23.837812 \nL 245.768548 23.837812 \nL 244.649394 23.837812 \nL 243.53024 23.837812 \nL 242.411087 23.837812 \nL 241.291933 23.837812 \nL 240.172779 23.837812 \nL 239.053625 23.837812 \nL 237.934471 23.837812 \nL 236.815317 23.837812 \nL 235.696163 23.837812 \nL 234.57701 23.837812 \nL 233.457856 23.837812 \nL 232.338702 23.837812 \nL 231.219548 23.837812 \nL 230.100394 23.837812 \nL 228.98124 23.837812 \nL 227.862087 23.837812 \nL 226.742933 23.837812 \nL 225.623779 23.837812 \nL 224.504625 23.837812 \nL 223.385471 23.837812 \nL 222.266317 23.837812 \nL 221.147163 23.837812 \nL 220.02801 23.837812 \nL 218.908856 23.837812 \nL 217.789702 23.837812 \nL 216.670548 23.837812 \nL 215.551394 23.837812 \nL 214.43224 23.837812 \nL 213.313087 23.837812 \nL 212.193933 23.837812 \nL 211.074779 23.837812 \nL 209.955625 23.837812 \nL 208.836471 23.837812 \nL 207.717317 23.837812 \nL 206.598163 23.837812 \nL 205.47901 23.837812 \nL 204.359856 23.837812 \nL 203.240702 23.837812 \nL 202.121548 23.837812 \nL 201.002394 23.837812 \nL 199.88324 23.837812 \nL 198.764087 23.837812 \nL 197.644933 23.837812 \nL 196.525779 23.837812 \nL 195.406625 23.837812 \nL 194.287471 23.837812 \nL 193.168317 23.837812 \nL 192.049163 23.837812 \nL 190.93001 23.837812 \nL 189.810856 23.837812 \nL 188.691702 23.837812 \nL 187.572548 23.837812 \nL 186.453394 23.837812 \nL 185.33424 23.837812 \nL 184.215087 23.837812 \nL 183.095933 23.837812 \nL 181.976779 23.837812 \nL 180.857625 23.837812 \nL 179.738471 23.837812 \nL 178.619317 23.837812 \nL 177.500163 23.837812 \nL 176.38101 23.837812 \nL 175.261856 23.837812 \nL 174.142702 23.837812 \nL 173.023548 23.837812 \nL 171.904394 23.837812 \nL 170.78524 23.837812 \nL 169.666087 23.837812 \nL 168.546933 23.837812 \nL 167.427779 23.837812 \nL 166.308625 23.837812 \nL 165.189471 23.837812 \nL 164.070317 23.837812 \nL 162.951163 23.837812 \nL 161.83201 23.837812 \nL 160.712856 23.837812 \nL 159.593702 23.837812 \nL 158.474548 23.837812 \nL 157.355394 23.837812 \nL 156.23624 23.837812 \nL 155.117087 23.837812 \nL 153.997933 23.837812 \nL 152.878779 23.837812 \nL 151.759625 23.837812 \nL 150.640471 23.837812 \nL 149.521317 23.837812 \nL 148.402163 23.837812 \nL 147.28301 23.837812 \nL 146.163856 23.837812 \nL 145.044702 23.837812 \nL 143.925548 23.837812 \nL 142.806394 23.837812 \nL 141.68724 23.837812 \nL 140.568087 23.837812 \nL 139.448933 23.837812 \nL 138.329779 23.837812 \nL 137.210625 23.837812 \nL 136.091471 23.837812 \nL 134.972317 23.837812 \nL 133.853163 23.837812 \nL 132.73401 23.837812 \nL 131.614856 23.837812 \nL 130.495702 23.837812 \nL 129.376548 23.837812 \nL 128.257394 23.837812 \nL 127.13824 23.837812 \nL 126.019087 23.837812 \nL 124.899933 23.837812 \nL 123.780779 23.837812 \nL 122.661625 23.837812 \nL 121.542471 23.837812 \nL 120.423317 23.837812 \nL 119.304163 23.837812 \nL 118.18501 23.837812 \nL 117.065856 23.837812 \nL 115.946702 23.837812 \nL 114.827548 23.837812 \nL 113.708394 23.837812 \nL 112.58924 23.837812 \nL 111.470087 23.837812 \nL 110.350933 23.837812 \nL 109.231779 23.837812 \nL 108.112625 23.837812 \nL 106.993471 23.837812 \nL 105.874317 23.837812 \nL 104.755163 23.837812 \nL 103.63601 23.837812 \nL 102.516856 23.837812 \nL 101.397702 23.837812 \nL 100.278548 23.837812 \nL 99.159394 23.837812 \nL 98.04024 23.837812 \nL 96.921087 23.837812 \nL 95.801933 23.837812 \nL 94.682779 23.837812 \nL 93.563625 23.837812 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46.949578 \nL 64.465625 48.394064 \nL 64.465625 49.838549 \nL 64.465625 51.283034 \nL 64.465625 52.72752 \nL 64.465625 54.172005 \nL 64.465625 55.61649 \nL 64.465625 57.060976 \nL 64.465625 58.505461 \nL 64.465625 59.949946 \nL 64.465625 61.394432 \nL 64.465625 62.838917 \nL 64.465625 64.283402 \nL 64.465625 65.727888 \nL 64.465625 67.172373 \nL 64.465625 68.616859 \nL 64.465625 70.061344 \nL 64.465625 71.505829 \nL 64.465625 72.950315 \nL 64.465625 74.3948 \nL 64.465625 75.839285 \nL 64.465625 77.283771 \nL 64.465625 78.728256 \nL 64.465625 80.172741 \nL 64.465625 81.617227 \nL 64.465625 83.061712 \nL 64.465625 84.506197 \nL 64.465625 85.950683 \nL 64.465625 87.395168 \nL 64.465625 88.839654 \nL 64.465625 90.284139 \nL 64.465625 91.728624 \nL 64.465625 93.17311 \nL 64.465625 94.617595 \nL 64.465625 96.06208 \nL 64.465625 97.506566 \nL 64.465625 98.951051 \nL 64.465625 100.395536 \nL 64.465625 101.840022 \nL 64.465625 103.284507 \nL 64.465625 104.728992 \nL 64.465625 106.173478 \nL 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64.465625 165.397377 \nL 64.465625 166.841863 \nL 64.465625 168.286348 \nL 64.465625 169.730833 \nL 64.465625 171.175319 \nL 64.465625 172.619804 \nL 64.465625 174.064289 \nL 64.465625 175.508775 \nL 64.465625 176.95326 \nL 64.465625 178.397746 \nL 64.465625 179.842231 \nL 64.465625 181.286716 \nL 64.465625 182.731202 \nL 64.465625 184.175687 \nL 64.465625 185.620172 \nL 64.465625 187.064658 \nL 64.465625 188.509143 \nL 64.465625 189.953628 \nL 64.465625 191.398114 \nL 64.465625 192.842599 \nL 64.465625 194.287084 \nL 64.465625 195.73157 \nL 64.465625 197.176055 \nL 64.465625 198.620541 \nL 64.465625 200.065026 \nL 64.465625 201.509511 \nL 64.465625 202.953997 \nL 64.465625 204.398482 \nL 64.465625 205.842967 \nL 64.465625 207.287453 \nL 64.465625 208.731938 \nL 64.465625 210.176423 \nL 64.465625 211.620909 \nL 64.465625 213.065394 \nL 64.465625 214.509879 \nL 64.465625 215.954365 \nL 64.465625 217.39885 \nL 64.465625 218.843336 \nL 64.465625 220.287821 \nL 64.465625 221.732306 \nL 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            "text/plain": [
              "\u003cFigure size 600x600 with 1 Axes\u003e"
            ]
          },
          "metadata": {
            "tags": []
          },
          "output_type": "display_data"
        }
      ],
      "source": [
        "plt.contourf(W_var, b_var, c_star(W_var, b_var) \u003e 0.999, \n",
        "             levels=3, \n",
        "             colors=[[1.0, 0.89, 0.811], [0.85, 0.85, 1]])\n",
        "plt.contourf(W_var, b_var, \n",
        "             jnp.abs(chi_1(W_var, b_var) - 1) \u003c 0.003, \n",
        "             levels=[0.5, 1], \n",
        "             colors=[[0, 0, 0]])\n",
        "\n",
        "plt.title('Phase diagram in terms of weight and bias variance', fontsize=14)\n",
        "\n",
        "format_plot('$\\\\sigma_w^2$', '$\\\\sigma_b^2$')\n",
        "finalize_plot((1, 1))"
      ]
    }
  ],
  "metadata": {
    "accelerator": "GPU",
    "colab": {
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        "aWMU1YkS5lPB"
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        "build_target": "//learning/deepmind/dm_python:dm_notebook3",
        "kind": "private"
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      "name": "Phase Diagram.ipynb",
      "provenance": [
        {
          "file_id": "/piper/depot/google3/third_party/py/neural_tangents/notebooks/phase_diagram.ipynb",
          "timestamp": 1586275485195
        },
        {
          "file_id": "1nES4ZNcUpN9Hoj1BuIGY_ZBN9Ek2sC_4",
          "timestamp": 1579895439402
        }
      ]
    },
    "kernelspec": {
      "display_name": "Python 3",
      "name": "python3"
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  "nbformat": 4,
  "nbformat_minor": 0
}
